Write the equation for a parabola with a focus at $(7,-6)$ and a directrix at $x=5$. $x=$
Solution: The strategy A parabola is defined as the set of all points that are the same distance away from a point (the focus) and a line (the directrix). Let $(x,y)$ be a point on the parabola. Then the distance between $(x,y)$ and the focus, $(7,-6)$, is equal to the distance between $(x,y)$ and the directrix, $x=5$. Once we find these distances, we can equate them in order to derive the equation of our parabola. Finding the distances from $(x,y)$ to the focus and the directrix The distance between $(x,y)$ and $(7,-6)$ is $\sqrt{(x-7)^2+(y+6)^2}$. [How did we find that?] Similarly, the distance between $(x,y)$ and the line $x=5$ is $\sqrt{(x-5)^2}$. [How did we know that?] Deriving the formula by equating the distances $\begin{aligned} \sqrt{(x-5)^2} &= \sqrt{(x-7)^2+(y+6)^2} \\\\ (x-5)^2 &= (x-7)^2+(y+6)^2 \\\\ {x^2}-10x{+25} &= {x^2}{-14x}+49+(y+6)^2\\\\ -10x{+14x}&=(y+6)^2+49{-25} \\\\ 4x&=(y+6)^2+24 \\\\ x&=\dfrac{(y+6)^2}{4}+6\end{aligned}$ The answer The equation of our parabola is $x=\dfrac{(y+6)^2}{4}+6$. Here is the graph of our parabola. As expected, the distance between a point on the parabola, $(x,y)$, and the focus is the same as the distance between $(x,y)$ and the directrix. ${2}$ ${4}$ ${6}$ ${8}$ ${10}$ ${12}$ ${14}$ ${16}$ ${\llap{-}2}$ ${\llap{-}4}$ ${2}$ ${4}$ ${6}$ ${8}$ ${10}$ ${\llap{-}2}$ ${\llap{-}4}$ ${\llap{-}6}$ ${\llap{-}8}$ ${\llap{-}10}$ $y$ $x$ ${(x,y)}$